INVESTIGATIONS ON ACCURATE ANALYSIS OF MICROSTRIP REFLECTARRAYS

M. Zhou1,2, S. B. Sørensen1, O. S. Kim2, S. Pivnenko2, and G. Toso3

1TICRA, Læderstræde 34, DK-1201 Copenhagen, Denmark2Department of Electrical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

3ESA-ESTEC, Keplerlann 1, PB 299, 2200 AG Noordwijk, The Netherlands

ABSTRACT

An investigation on accurate analysis of microstrip re-flectarrays is presented. Sources of error in reflectarrayanalysis are examined and solutions to these issues areproposed. The focus is on two sources of error, namelythe determination of the equivalent currents to calculatethe radiation pattern, and the inaccurate mutual couplingbetween array elements due to the lack of periodicity. Toserve as reference, two offset reflectarray antennas havebeen designed, manufactured and measured at the DTU-ESA Spherical Near-Field Antenna Test Facility. Com-parisons of simulated and measured data are presentedto verify and demonstrate the improved results using theproposed solutions.

Key words: Microstrip reflectarrays; accurate antennaanalysis; method of moments (MoM), antenna radiationpattern, horn antennas, measurements.

1. INTRODUCTION

Microstrip reflectarrays are becoming viable alternativesto reflector antennas for satellite applications and havebeen the subject of increasing research interest [1–4]. Toobtain high-gain performance for satellite applications,the electric size of reflectarrays is usually very large,and therefore an efficient and accurate analysis is a chal-lenging task. The commonly adopted analysis method isbased on the spectral domain Method of Moments (SD-MoM) assuming local periodicity (LP), that is, an indi-vidual array element is embedded in an infinite array ofidentical elements [5]. This approach has been demon-strated to be efficient for reflectarrays made of varying-sized patches, and many advanced reflectarrays have beendesigned using this technique [3]. However, reflectarraysare aperiodic by nature and the local periodicity assump-tion gives rise to discrepancies between simulated andmeasured radiation patterns. Efficient full-wave simula-tion techniques have been applied on entire reflectarraysfor accurate determination of the currents on the arrayelements [6–9]. However, discrepancies between simu-lations and measurements can still be observed, and theincrease in computation time makes the methods unaf-fordable for optimization processes. For space applica-tions, where the accuracy demands are high, an efficient

and accurate analysis method is important to precisely de-termine the radiation properties of reflectarrays, and it isessential for optimization purposes. The objective of thiswork is to understand the sources of error in reflectarrayanalysis and to propose solutions to these.

This paper is organized as follows. Section 2 describessome of the sources of error in reflectarray analysis. Thebenchmark antennas are described in Section 3. In Sec-tion 4, simulations are compared with measurements, andconclusions are given in Section 5.

2. SOURCES OF ERROR

In reflectarray analysis, several factors can give rise toerrors, e.g. the representation of the incident field, thechoice of basis functions, the technique used to calculatethe radiation patterns, the lack of periodicity, the trun-cation of the ground plane, scattering from the supportstructures etc. All these factors should be accounted forcorrectly in order to obtain an accurate analysis.

2.1. Representation of the Incident FieldIn the SDMoM computations, each array element is as-sumed to be illuminated by a locally plane wave. Thepattern of the feed is usually approximated by a far fieldmodel using a cosq(θ) function or a Gaussian beam [1,Sec. 3.8], and is used to compute the polarization of theincident plane wave on each array element. This approxi-mation is usually inaccurate due to the idealized feed pat-tern. The analysis accuracy can be improved by usingthe real pattern of the feed obtained by either measure-ments [10] or commercially available simulation toolse.g. CHAMP [11]. However, the assumption of planewave incidence in the SDMoM is only valid if the reflec-tarray surface is located sufficiently far away from thefeed. Alternatively, a plane wave expansion of the feedradiation over the reflectarray surface can be computed.The SDMoM analysis is then performed for each planewave and subsequently added to yield the final result. Inthis approach, the representation of the incident field isexact, however at the cost of computation time. For thereflectarray antennas to be described in Section 3, it wasfound that the assumption of plane wave incidence is asufficient representation of the incident field.

2.2. Choice of Basis Functions

The choice of basis functions in the SDMoM computa-tions is important. Roof-tops and entire domain trigono-metric basis functions are popular choices. However, forresonant microstrip array elements, the convergence ofthe SDMoM solution becomes poor and in certain casesconvergence is never achieved using these basis func-tions. Consequently, singular basis functions with correctedge conditions are required for accurate characterizationof the array elements [12]. However, singular basis func-tions require additional Floquet modes to achieve conver-gence, thus increasing the total computation time. In thispaper, higher order hierarchical Legendre basis functionsas described in [13] are used. These basis functions canbe applied to any non-canonical element types, and at thesame time yield results that are identical to those obtainedwith singular basis functions using less computation time[14].

2.3. Aperiodicity

Reflectarrays are inherently aperiodic due to the need tocompensate for the spatial phase delay from the feed.Thus the periodicity assumption in the LP approach cangive inaccurate results.

One way to reduce these errors is to use the ”SurroundedElement Approach” (SEA) proposed in [15]. The anal-ysis in SEA is based on a finite approach where no pe-riodicity is applied. It includes the actual neighboringelements that surrounds the element under consideration,thus accounting for the mutual coupling more accurately.In [15], the analysis was based on a FDTD implementa-tion assuming plane wave incidence, and many neighbor-ing elements were required to obtain an improved result.Even though the reported computation time were in termsof hours, we find this technique interesting and promis-ing. Therefore, an integral equation (IE) formulation us-ing a spatial dyadic Green’s function (DGF) [16] is cur-rently being implemented. Using an IE formulation, theplane wave incidence assumption is avoided and the to-tal computation time may be reduced. Preliminary resultsindicate that the SEA is especially accurate in predictingthe radiation pattern of the cross-polarization component.

Another approach, which is a combination of the LP ap-proach and the SEA, is introduced in [14], the ”ExtendedLocal Periodicity” (ELP) approach. Similar to the LP ap-proach, the ELP approach is also based on periodicity, butthe periodicity is applied on an extended unit cell whichincludes the actual 8 neighboring elements that surroundthe element under consideration. SDMoM is applied tothe extended unit cell and the unknown currents on theelement under consideration is determined. This is re-peated for the next element with the extended unit cellnow including the new element under consideration andits 8 neighbors. The inclusion of the nearest surroundingneighbors increases the total computation time, since thenumber of basis functions is larger and additional Flo-quet modes are required. While an analysis of a realisticreflectarray takes a couple of seconds using the LP ap-

proach, the ELP approach requires 40-60 minutes. Thisincrease is significant and must be decreased if the ap-proach is to be used for optimization purposes. Acceler-ation techniques on this matter can be found in the litera-ture [17, 18] and are currently being investigated.

Recently, there have been interests in completely aperi-odic reflectarrays that aim to exploit all available degreesof freedom of the array elements e.g. element positionsand orientations [19, 20]. The array elements are locatedin irregular grids and the periodic assumption in the LPapproach is inaccurate. It is expected that the SEA andthe ELP approach are better candidates for analysis ofsuch reflectarrays.

2.4. Truncation Effects

Due to the periodicity assumption in the LP and ELP ap-proaches, the truncation of the ground plane is not takeninto account in the determination of the currents on thearray elements. Nor is it accounted for in the IE for-mulations since the spatial DGF assumes infinite groundplane. An efficient and accurate solution to this issueshould be investigated.

2.5. Radiation Pattern Calculations

In the literature on accurate analysis of reflectarrays, themain focus has been on the determination of the currentson the array elements, and the technique used to calculatethe radiation pattern has received less attention. However,the latter is equally important.

The conventional technique, which will be referred to astechnique I, to calculate the radiation pattern is to in-voke the field equivalence principle to calculate equiva-lent electric and magnetic surface currents in the plane ofthe array elements [21, 22]. The equivalent electric andmagnetic currents are defined as

JS =∑i

Ji =∑i

z ×Hi, (1)

MS =∑i

Mi = −∑i

z ×Ei, (2)

where index i runs over all array elements. The elec-tric and magnetic fields, Ei and Hi, respectively, are ap-proximated using the fundamental Floquet mode from thediscrete spectrum in the SDMoM simulations. By inte-grating the equivalent currents, the far field radiation pat-tern can be determined. There are several disadvantageswith this technique. Firstly, the ground plane in reflectar-rays is often extended at the edges for practical reasons[22, 23]. The extended ground plane contributes to the farfield radiation and is not accounted for in this technique.This can be circumvented by placing unit cells with noelements at the edges to cover the area of the extendedground plane. However, this procedure is rather imprac-tical and not suitable for commercial codes. Secondly,equivalent currents are discontinuous at cell boundaries.

(a) Antenna I (b) Antenna II

Figure 1. Reflectarray mask for the antenna I and II.

These discontinuities can contribute to phase and ampli-tude errors in the equivalent currents, thus resulting inan erroneous relation between the equivalent electric andmagnetic currents. As a result, the radiation in the backhemisphere can be wrong [24].

To avoid the issues associated with technique I, a noveltechnique as described in [24] can be used. This tech-nique will be referred to as technique II and it also uti-lizes the field equivalence principle. Instead of calculat-ing the equivalent currents using the discrete spectrumfrom the SDMoM simulations, a continuous spectrum isemployed. In this technique, the extended ground plane isautomatically accounted for. Additionally, no discontinu-ities are created in the equivalent currents, and the equiv-alent electric and magnetic currents are correctly relatedthrough the continuous spectrum, thus enabling an accu-rate calculation of the radiation pattern on the entire farfield sphere.

3. BENCHMARK ANTENNAS

To serve as benchmark cases, two offset microstrip re-flectarray antennas were designed. The designs are fo-cused on the aforementioned sources of error and are in-tended to exaggerate these sources such that they can beseparated. The radiation patterns of the antennas weremeasured at the DTU-ESA Spherical Near-Field AntennaTest Facility [25] and serve as reference solutions.

The reflectarray in Figure 1a is designed to exaggerate thetruncation effects. To do so, a strong edge illumination isrequired and the aperiodicity effects must be reduced. Tothis end, a smooth patch variation is obtained by reduc-ing the spatial phase delay by placing the feed far fromthe reflectarray surface, and by directing a pencil beamtowards the specular direction.

The reflectarray in Figure 1b is designed to exaggeratethe aperiodicity in reflectarrays by having a pencil beamtowards θ = 35◦ and φ = 135◦ in the coordinate system

x

z

dfθi

dx

(a)

x

y

φi

dx

dy

(b)

Figure 2. Geometrical parameters of the reflectarray (a)the xz-plane (b) the xy-plane.

in Figure 2. To ensure that the truncation effects are neg-ligible, the feed is located close to the reflectarray surfacesuch that a low edge illumination can be achieved with ahigh gain feed.

For both antennas, the substrate is Rogers RO4350B witha relative permittivity of εr = 3.66 and a loss tangentof tanδ = 0.0037. The ground plane and substrate forboth antennas are slightly extended at the edges as seen inFigure 1. The geometrical parameters of the reflectarraysare shown in Figure 2 and summarized in Table 1.

Two horn antennas were used as feed, a corrugated horn(Figure 3a), and a Potter horn (Figure 3b). Each hornwas used on both antennas giving a total of 4 differentreflectarray configurations. The corrugated horn has at10GHz a taper of −17.5 dB at 30◦ whereas the Potterhorn has a taper of −7 dB at 30◦.

The reflectarrays and their support structures were man-ufactured at the Technical University of Denmark. Forthe measurements, see Figure 4, the estimated 1σ uncer-tainty for the peak directivity is 0.07 dB. In addition tothe reflectarray measurements, the feed horns were alsomeasured and the measured data was used to representthe incident field on the reflectarray surface in the analy-sis.

Table 1. Benchmark Antenna Data

Frequency 10GHzNumber of elements 30×30Reflectarray dimensions 435mm× 435mmSubstrate thickness 0.762mmRelative permittivity (εr) 3.66Loss tangent (tan δ) 0.0037Antenna IFeed distance: df = 0.6mFeed orientation: θi = 30◦, φi = 0◦

Main beam direction: θ = −30◦, φ = 0◦

Antenna IIFeed distance: df = 0.35mFeed orientation: θi = 45◦, φi = 0◦

Main beam direction: θ = 35◦, φ = 135◦

(a) (b)

Figure 3. The two horn antennas used in the benchmarkcases, (a) the corrugated horn, (b) the Potter horn.

These reflectarray antennas and their associated radia-tion patterns constitute a scientific contribution in itself,since the results currently available in the literature arenot completely specified in terms of geometrical parame-ters or feed radiation pattern, or the accuracy of the mea-surements. In addition, the available results are for com-plex full-size reflectarrays where many sources of errorsare present at the same time and cannot be distinguished.

4. SIMULATIONS VS. MEASUREMENTS

Radiation patterns obtained at 10GHz by measurementsand simulations are presented in this section. To accountfor the presence of the support structures, the scatteringfrom the struts is included in the analysis using the MoMadd-on in GRASP [26]. All radiation patterns are de-picted in a coordinate system defined with the z-axis di-rected towards the main beam direction.

4.1. Radiation Pattern CalculationsTo demonstrate the accuracy of the aforementioned tech-niques to calculate the radiation pattern, technique I andII are used to compute the radiation patterns for antennasI and II, and compared with measurements in Figure 5and Figure 6. For these results, the currents on the arrayelements are determined using the LP approach. To illus-trate the effect of the extended substrate, this contributionis not included in technique I.

(a) Antenna I with the corrugated horn

(b) Antenna II with the Potter horn

Figure 4. Benchmark antennas at the DTU-ESA Spheri-cal Near-Field Antenna Test Facility.

The far field co-polar radiation patterns for antenna I areshown in Figure 5. The Potter horn is used in this caseto give a strong edge illumination, −7 to −1 dB. It isseen that the improvement using technique II is clear, es-pecially in Figure 5b where the simulated and measuredpattern practically coincide. The discrepancies observedin Figure 5a around θ = 45◦ are due to the blockage ofthe measurement tower.

The far field co-polar radiation patterns for antenna IIare depicted in Figure 6. A low edge illumination of ap-proximately −25 to −12 dB is achieved in this case us-ing the corrugated horn as feed. Even in this case wherethe edges are weakly illuminated, the enhancement of theanalysis accuracy is apparent. The side lobes are muchbetter predicted using technique II compared to techniqueI. The discrepancies are mainly attributed to the strongaperiodicity.

The improvements in Figure 5 and Figure 6 using tech-nique II are due to the inclusion of the extended groundplane. It is shown in [24] that a similar accuracy in theforward hemisphere can be achieved with technique I ifempty unit cells are placed at the edges to cover the areaextended by the substrate. Since very good agreementscan be achieved when the aperiodicity effects are mini-mized, it can be concluded that the error introduced bythe truncation of the ground plane is of small importance

−90 −45 0 45 90−20

−10

0

10

20

30

θ [◦]

Dir

ectiv

ity[d

Bi]

φ = 0◦

Technique ITechnique IIMeasurements

(a)

−90 −45 0 45 90−20

−10

0

10

20

30

θ [◦]

φ = 90◦

Technique ITechnique IIMeasurements

(b)

Figure 5. Comparison of radiation pattern calculated using technique I and II with measurements for antenna I.

−90 −45 0 45 90−20

−10

0

10

20

30

θ [◦]

Dir

ectiv

ity[d

Bi]

φ = 0◦

Technique ITechnique IIMeasurements

(a)

−90 −45 0 45 90−20

−10

0

10

20

30

θ [◦]

φ = 135◦

Technique ITechnique IIMeasurements

(b)

Figure 6. Comparison of radiation pattern calculated using technique I and II with measurements for antenna II.

−180 −135 −90 −45 0 45 90 135 180−30

−20

−10

0

10

20

30

θ [◦]

Dir

ectiv

ity[d

Bi]

φ = 135◦

Technique ITechnique IIMeasurements

Figure 7. The results from Fig. 6b shown also in the backhemisphere.

for the determination of the currents on the array ele-ments, but has to be accounted for in the calculation ofthe far field radiation pattern.

With respect to the radiation in the back hemisphere,technique I can yield wrong results as shown in Figure 7.In this figure, the results from Figure 6b are shown also inthe back hemisphere. It is seen that in the direction of themain beam’s image around θ = 145◦, an erroneous beamis predicted by technique I. This is caused by the incorrectrelation between the equivalent electric and magnetic cur-rents in technique I as previously mentioned. This issueis avoided using technique II since the equivalent currentsare correctly related to each other, thus yielding good re-sults in the entire far field sphere.

4.2. AperiodicityTo illustrate the aperiodicity effects, results for antennaII computed using the LP and ELP approaches are com-

−180 −135 −90 −45 0 45 90 135 180−20

−10

0

10

20

30

θ [◦]

Dir

ectiv

ity[d

Bi]

φ = 0◦

LPELPMeasurements

(a)

−180 −135 −90 −45 0 45 90 135 180−20

−10

0

10

20

30

θ [◦]

Dir

ectiv

ity[d

Bi]

φ = 90◦

LPELPMeasurements

(b)

Figure 8. Comparison of simulations using LP and ELP, and measurements for antenna II. The Potter horn is used asfeed.

−10 −5 0 5 10−10

0

10

20

30

θ [◦]

Dir

ectiv

ity[d

Bi]

φ = 0◦

LPELPMeasurements

Figure 9. Close up of the comparison shown in Fig. 8a.

pared with the measurements and shown in Figure 8.Technique II is used to calculate the radiation patterns,and the Potter horn is used as feed yielding an edge illu-mination of approximately −20 to −5 dB.

It is seen that the overall accuracy over the entire farfield sphere is good, also for the LP case. This is dueto the technique used to calculate the radiation patternand the inclusion of the radiation from the support struc-tures. However, it is observed that the ELP approach isgenerally more accurate e.g. −90 < θ < −40 in Fig-ure 8b. To see the improvements more clearly, a closeup of the radiation pattern from Figure 8a is depicted inFigure 9. It is apparent that the accuracy is enhancedusing the ELP approach. The peak directivity is mea-sured to Dmeas = 26.1 dBi. The LP approach predictsDmeas = 26.8 dBi whereas ELP yields an improvedvalue of Dmeas = 25.9 dBi. For the far field cross-polar radiation patterns, the results obtained using the LPand ELP approaches are rather similar and improvements

were not observed.

5. CONCLUSIONS

An investigation on accurate analysis of microstrip reflec-tarrays has been carried out and is presented in this paper.Several sources of error in reflectarray analysis have beenexamined, e.g. the representation of the incident field,the choice of basis functions, aperiodicity, and truncationeffects, as well as the technique used to calculate the ra-diation pattern. To serve as benchmark cases, two offsetreflectarray antennas have been designed, manufacturedand measured at the DTU-ESA Spherical Near-Field An-tenna Test Facility. The two designs aim to exaggerate theerrors introduced by the truncation of the ground plane,and the lack of periodicity, respectively. Comparisons ofsimulated and measured radiation patterns show that theerrors introduced by the truncation of the ground planeare of small importance for the determination of the cur-rents on the array elements and can thus be neglected.The errors introduced by the aperiodicity on the otherhand can be significant and the conventional local period-icity assumption can be inaccurate. To circumvent this,the extended local periodicity approach can be utilized.In addition, the comparisons shown that the choice of thetechnique to calculate the radiation pattern is very impor-tant with respect to the analysis accuracy. The finite sub-strate and ground plane size of the reflectarray must betaken into account and techniques that neglect this giveinaccurate results.

The results presented in this paper show that accurateanalysis of reflectarrays can be achieved if the sources oferror are treated correctly. However, several challengesremain and are presently being investigated.

ACKNOWLEDGMENTS

This work is supported by the European Space Agency(ESA) within the ESTEC contract No.4000101041.

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